In this paper, we study a family of take-away games called α-TAG, parametrized by a real number α ≥ 1. We show that for any given α, there is a half-open interval I_α containing α such that the set of losing positions for α-TAG is the same as the set of losing positions for β-TAG if and only if β ∈ I_α. We then end with some results and conjectures on the nature of these intervals.